**1. Introduction**

Low-pressure carburization especially combined with high-pressure gas quenching is a new technology that allows accurate control over the carbide morphology, size, distribution, surface carbon content, and layer depth [1–3]. It also offers energy savings and environmental protection and is being applied to various gear steels. However, for high alloy steel, it is easy to exceed the standard of carburized layer carbide after carburizing. The main reason is the accuracy of carbon flux in the process of formulation. Carbon flux represents the mass of material passing through a unit area perpendicular to diffusion direction per unit time. The value of carbon flux is related to factors such as the material composition, temperature, carburizing pressure, and catalyst [4,5]. During carburization, the gas contacting the steel surface undergoes a chemical reaction, but the growth in mass cannot be measured in real time, rendering it difficult to measure the carbon flux. Hwang et al. [6,7] studied the effect of material composition and surface oxidation on carbon flux, and Kula et al. [8] studied the effect of carbides on carbon flux at the surface. Furthermore, Karabelchtchikova and Sisson [9] proposed an improved integration method to determine the carbon flux in the gas boundary layer in austenite, stating that the carbon flux changes with both time and carbon concentration on the workpiece surface. Zajusz et al. [10] optimized the carbon transfer coefficient in their research. They analyzed the effects of pressure, reaction rate, and alloy elements through extensive experiments, thereby obtaining the effect of process conditions on carbon flux, which achieved good results when applied to the model of carburizing. Lowell [11] studied the effect of workpiece surface roughness and the type of contaminant on the carbon flux of low-pressure vacuum carburizing surface. Gorockiewicz et al. [12,13] found that in the boost stage of carburization, carbon atoms are first adsorbed by the workpiece surface to form crystals, and then diffused as a carbon source. Su [14] proposed a saturated carburizing model for carbide migration and calculated the variation of average carbon flux with carburizing time. However, these studies mainly focus on low carbon steel and low alloy steel. For high alloy steel, it is difficult to control the surface carbides. The carbon flux is affected by many factors, so it is hard to determine the carbon flux accurately [15,16].

For low-pressure vacuum carburizing, the carbon potential of the atmosphere cannot be measured or controlled. The measurement of carbon flux is also difficult and there is no suitable method for on-line continuous testing. Philippe et al. [17] experimentally measured the change in carbon flux using a thin saturated carburizing iron foil with a carburizing atmosphere on one side and a decarburizing atmosphere on the other side. Therein, the carbon flux was determined by measuring the carbon flow rate in a decarburizing atmosphere. Owing to the short duration of the boost stage, the actual measurement precision and response time were difficult to control. Therefore, the application of this method to low-pressure vacuum carburization in real-world applications requires further research.

Oriental Furnace (Japan) and Dowa Holdings proposed a method to determine the carbon concentration using a high-temperature hydrogen probe [18] to analyze the hydrogen content in a gas mixture. Khan et al. [19] studied the cracking of propane during low-pressure vacuum carburization for carburization control. Furthermore, Yada et al. [20] and Makino et al. [21] numerically simulated and tested the carburization process and detected the decomposition of acetylene using the volume fraction of hydrogen in the gas mixture to calculate the carbon potential in the furnace. Thus far, this technique has not been successfully applied in industrial applications, and the values of relevant parameters remain confidential.

Currently, majority carbon flux determination methods utilize an experimentally obtained average value, without adequately considering the effects of process parameters such as carburizing temperature and pressure and material composition; this results in a relatively large experimental error. In practice, carbon flux continuously changes in the carburizing process; hence, most carbon flux values exhibit large deviation in real-life applications, particularly for high alloy materials. To the best of our knowledge, there has been no report on mathematical modeling related to this issue.

In the present study, a comparison of the carburized layer organizations obtained by the average and segmented average carbon flux models has been given. It is found that carbide network is more easily to form on the surface-carburized layer with an average carbon flux model. To accurately control the carburizing process, a systematic study with each model using different materials (12Cr2Ni4A, 16Cr3NiWMoVNbE, and 18Cr2Ni4WA represent different initial carbon concentrations and different alloy compositions), carburizing temperatures, and carburizing pressures is conducted to determine the effect of these conditions on carbon flux.

